# Thread: Determining the countability of a set of function.

1. ## Determining the countability of a set of function.

Determine whether or not the set D of all functions $f:\mathbb{Z}_+\to\mathbb{Z}_+{$ is countable. Justify your answer.

If anyone could give me some hints I could use to get started on this question I would really appreciate it. I have been trying to use the fact that D is a subset of the power set of $\mathbb{Z}_+\times\mathbb{Z}_+$ but is have not really been able to get anywhere with that.

2. The set of all functions $f:\mathbb{Z}^+\to \{0,1\}=2^{\mathbb{Z}^+ }$ which in equipotent to $\mathscr{P}\left(\mathbb{Z}^+ \right)$, the power set of $\mathbb{Z}^+$.

Is it not true that $\left| {\mathbb{Z}^ + } \right| < \left| {\mathscr{P}\left( {\mathbb{Z}^ + } \right)} \right|?$